Nonwandering

Nonwandering is a term used in mathematics, particularly in the study of dynamical systems and ergodic theory, to describe the behavior of points within a phase space. A point is considered nonwandering if every neighborhood around it is visited infinitely often by the system's trajectories. In simpler terms, if you start a point in a particular region and let the system evolve, a nonwandering point will eventually return to that region, and continue to do so, infinitely many times.

The set of all nonwandering points for a dynamical system forms a closed invariant set. This set is crucial because it often contains the essential dynamics of the system. For example, in a system with chaotic behavior, the nonwandering set might encompass the strange attractor. Conversely, if a point is not nonwandering, it is called a wandering point. This means there exists some neighborhood around it that the trajectory will eventually leave and never return to.

The concept of nonwandering points helps to classify different types of dynamical systems and understand their long-term behavior. It is a fundamental idea for analyzing the stability and recurrence properties of solutions to differential equations and discrete maps. The distinction between wandering and nonwandering points provides a way to separate predictable, repetitive behavior from transient or chaotic dynamics.